IP Maths Notes (Lower Sec, Year 1-2): 08) Trigonometry in Right Triangles

Trigonometry unlocks height and distance tasks across science and geography contexts. Become fluent with ratios, inverse functions, and diagram interpretation.

Learning targets

• Recall primary trigonometric ratios: \( \sin \theta = \frac{\text{opp}}{\text{hyp}} \), \( \cos \theta = \frac{\text{adj}}{\text{hyp}} \), \( \tan \theta = \frac{\text{opp}}{\text{adj}} \).
• Solve for sides and angles using calculator and exact surd values.
• Handle angles of elevation and depression with annotated diagrams.
• Apply the sine and cosine rules preview (basic awareness for oblique triangles).

1. Ratio recap

Label the triangle carefully: mark the angle of focus, and label opposite, adjacent, hypotenuse relative to it.

Memory aid: SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Worked example — Finding a side

Given right triangle with angle \( \theta = 37^\circ \) and hypotenuse 12 cm, find the opposite side.

\[ \sin 37^\circ = \frac{\text{opp}}{12} \implies \text{opp} = 12 \sin 37^\circ \approx 12 \times 0.601 = 7.21 \text{ cm}. \]

Worked example — Finding an angle

Opposite side 5 cm, adjacent side 8 cm. Find \( \theta \).

\[ \tan \theta = \frac{5}{8} \implies \theta = \tan^{-1}\left(\frac{5}{8}\right) \approx 32.0^\circ. \]

2. Exact values and surds

Memorise \( \sin 30^\circ = \frac{1}{2} \), \( \cos 45^\circ = \frac{\sqrt{2}}{2} \), \( \tan 60^\circ = \sqrt{3} \).

Use rationalisation to present answers cleanly: if \( \sin \theta = \frac{\sqrt{3}}{2} \), then \( \theta = 60^\circ \) or other equivalent angles depending on context.

3. Angle of elevation/depression

• Angle of elevation: measured upwards from horizontal.
• Angle of depression: measured downwards from observer to object.

Worked example — Two-point observation

A drone lifts off and reaches a point where the angle of elevation from a student 50 m away is \( 35^\circ \). What is the drone height?

\[ \tan 35^\circ = \frac{h}{50} \implies h = 50 \tan 35^\circ \approx 35.0 \text{ m}. \]

4. Bearings and composite paths

Bearings measured clockwise from north. Break vectors into horizontal/vertical components using cosine and sine, then recombine with Pythagoras.

Preview: sine and cosine rules

Although formally covered in upper sec, introduce the formulas for awareness:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \quad c^2 = a^2 + b^2 - 2ab \cos C. \]

Try it yourself

1. Find the hypotenuse of a right triangle with adjacent side 9 cm and angle \( 28^\circ \).
2. A ladder leans against a wall with its base 1.6 m from the wall and top 3.8 m above ground. Find the ladder length and angle of elevation.
3. Two observers stand 40 m apart on level ground. They observe the top of a pole with angles of elevation \( 32^\circ \) and \( 48^\circ \) respectively. Model the scenario and compute the pole height.
4. A yacht sails 12 km on a bearing of \( 040^\circ \) then 8 km on \( 130^\circ \). Calculate displacement from the starting point.

Wrap up with statistics at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-09-Data-Handling-and-Statistics.